In this thesis the free and forced vibrations of a single degree of freedom oscillator with a periodically time-varying mass have been studied. Linear and weakly non-linear oscillator equations have been considered. The forced vibrations of the oscillator are partly due to small masses which are T-periodically hitting and leaving the oscillator with T-periodic velocities. Since these small masses stay for some time on the oscillator surface the effective mass of the oscillator and the shape of the oscillator will periodically vary in time. The effect of a damping term (in the linear oscillator equation) on the solutions also has been considered. For the free vibrations the minimal damping rates have been computed for which the oscillator is always stable. Also cases with external, harmonic forcing have been investigated in detail for the linear oscillator equation, and interesting resonance conditions have been found. As simple model to describe the rain-wind induced oscillations of a cable, an initial value problem for an oscillator equation with a Rayleigh type of non-linearity has been studied.By applying a straight-forward perturbation method the problem has been solved approximately on a time-interval of length T. In all cases studied in this thesis initial value problems for oscillator equations have been formulated. The constructed solutions on a time-interval of length T or the approximations of the solutions on the same time-interval have been used to construct maps. By using these maps (i.e. by using a system of difference equations) the stability properties of the solutions have been determined. The instability regions in the parameter space have been computed partly analytically and partly numerically. Some phase-space figures for the weakly non-linear problem have been computed numerically to show a number of interesting bifurcations, and to show the rich dynamics of the problem.